Mathlib v4.30 source proposition Nat.coprime_or_dvd_of_prime

Identifier: F:ml430-nat-coprime-or-dvd-of-prime-65f47114
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Nat.coprime_or_dvd_of_prime` in the pinned Mathlib v4.30 source.

Formal statement

∀ {p : ℕ}, Nat.Prime p → ∀ (i : ℕ), p.Coprime i ∨ p ∣ i

Dependencies

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4 direct dependencies 6 direct dependents

Evidence

kernel-Nat.coprime_or_dvd_of_prime

Kind: kernel-term
Status: checked

Supports: ∀ {p : ℕ}, Nat.Prime p → ∀ (i : ℕ), p.Coprime i ∨ p ∣ i

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- coprime_or_dvd_of_prime 2>/dev/null | grep -Ec '^Nat\.coprime_or_dvd_of_prime[[:space:]]')" -ge 1

Evidence notes

`build_nat_prelude` admits `Nat.coprime_or_dvd_of_prime` through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type, so producing this row at all is a machine-checked proof. `nat_theorem_inventory` exits non-zero for a name that does not exist, and the `grep -c` count (tested `-ge 1`, not piped into `grep -q`) requires the admitted declaration to actually be printed. Decides `beq (gcd p i) one` via a `Bool.rec` case split (fully constructive, not classical excluded middle -- `Bool` has exactly two constructors): the `true` branch gives `Coprime p i` directly (`eq_of_beq_eq_true`); the `false` branch gives `Not (Coprime p i)` (`ne_of_beq_eq_false`), which `prime_dvd_iff_not_coprime`'s reverse direction converts to `dvd p i`.

footprint-Nat.coprime_or_dvd_of_prime

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the Nat prelude's trusted surface is empty

Checker command

cargo run -q -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free nat

Evidence notes

`nat_axiom_inventory --require-axiom-free nat` enumerates the built Nat environment and exits non-zero unless it admits no Axiom, Opaque or Quotient declaration (measured: axiom=0 opaque=0 quotient=0). A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds every individual theorem's footprint by [].

Provenance

{
  "date": "2026-08-18",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane nat-cascade",
  "source": "statement-only extraction of `Nat.coprime_or_dvd_of_prime` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.coprime_or_dvd_of_prime`",
      "where": "mathlib4 commit c5ea00351c28e24afc9f0f84379aa41082b1188f (v4.30.0)",
      "year": 2026,
      "attribution": "the proposition was read from the pinned statement-only inventory; the proof term and tactic trace were not consulted"
    }
  ]
}